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arXiv:1808.01585 · Physical review. B./Physical review. B

Critical exponents of the nonlinear sigma model on a Grassmann manifold U(N)/U(m)U(N−m) by the 1/N -expansion

Shan-Yue Wang, Da Wang, Qiang-Hua Wang

Published Apr 26, 2019• 5 citations
arXivHTMLPDF

Abstract

Motivated by the realization of $\mathrm{SU}(N)$ antiferromagnetism with the multirow representations in cold-atom physics, we studied its low-energy nonlinear sigma model defined on the Grassmann manifold $\text{U}(N)/\text{U}(m)\text{U}(N\ensuremath{-}m)$ using the complex projective presentation, which is a direct generalization of the widely studied ${\mathrm{CP}}^{N\ensuremath{-}1}$ model (corresponding to $m=1$). Using the large-$N$ expansion up to the first order of $1/N$ with fixed $m$, and in space dimension $2<d<4$, we obtain the critical exponents, including in particular the gauge-independent exponents that are directly relevant for experimental measurements. These exponents are found to depend only on $m/N$, indicating that a larger $m$ effectively reduces $N$ and thus causes stronger fluctuations about the saddle point.

DOI
10.1103/physrevb.99.165142

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Motivated by the realization of SU(N)\mathrm{SU}(N)SU(N) antiferromagnetism with the multirow representations in cold-atom physics, we studied its low-energy nonlinear sigma model defined on the Grassmann manifold U(N)/U(m)U(N\ensuremath−m)\text{U}(N)/\text{U}(m)\text{U}(N\ensuremath{-}m)U(N)/U(m)U(N\ensuremath−m) using the complex projective presentation, which is a direct generalization of the widely studied CPN\ensuremath−1{\mathrm{CP}}^{N\ensuremath{-}1}CPN\ensuremath−1 model (corresponding to m=1m=1m=1). Using the large-NNN expansion up to the first order of 1/N1/N1/N with fixed mmm, and in space dimension 2<d<42<d<42<d<4, we obtain the critical exponents, including in particular the gauge-independent exponents that are directly relevant for experimental measurements. These exponents are found to depend only on m/Nm/Nm/N, indicating that a larger mmm effectively reduces NNN and thus causes stronger fluctuations about the saddle point.
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We develop an optimized continuous-field quantum Monte Carlo (QMC) algorithm to investigate the projected SO(5) nonlinear sigma model with a Wess-Zumino-Witten term, which describes half-filled Dirac fermions in 2+1 space-time dimensions akin to graphene and Yukawa coupled to a quintuplet of compatible mass terms. Our algorithm reduces the computational complexity to O(βNqNϕ2)O(βN_{\mathbf{q}} N_ϕ^2)O(βNq​Nϕ2​), yielding a speedup of a factor of NϕN_ϕNϕ​ (the number of magnetic fluxes, i.e., system size) relative to prior works [1-4]. This advance enables us to simulate system sizes up to Nϕ=140N_ϕ=140Nϕ​=140 on the torus and Nϕ=59N_ϕ=59Nϕ​=59 on the sphere, far exceeding the maximum sizes previously accessed, and to map out the universal phase diagram of the model on both geometries. Most notably, we identify and characterize a critical point that separates an SO(5)-broken ordered phase at small coupling from an SO(5)-symmetric disordered phase at large coupling. The critical point becomes multicritical upon the inclusion of terms that break the SO(5) symmetry down to U(1)×SU(2)\mathrm{U}(1) \times \mathrm{SU}(2)U(1)×SU(2), relevant for the deconfined phase transition between Néel antiferromagnetic and valence-bond-solid orders in quantum magnets. Our finding of a multicritical point in the phase diagram of the SO(5) nonlinear sigma model with Wess-Zumino-Witten term resolves the long-standing open question of its global structure, and our QMC algorithm opens a new avenue for systematic studies of projected Hamiltonians, ranging from correlated flat bands to fractional quantum (anomalous) Hall systems.
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